a subgroup of , such that
for all .
The second condition says that for any such that for some integer and some , then there exists such that . In other words, if is divisible in by an integer, then it is divisible in by that same integer. Purity in abelian groups is a relative notion, and we denote to mean that is a pure subgroup of .
Examples. All groups mentioned below are abelian groups.
For any group, two trivial examples of pure subgroups are the trivial subgroup and the group itself.
Any divisible subgroup (http://planetmath.org/DivisibleGroup) or any direct summand of a group is pure.
If , , then .
If with and , then .
In , is an example of a subgroup that is not pure.
In general, if , where and .
Remark. This definition can be generalized to modules over commutative rings.
Definition. Let be a commutative ring and a short exact sequence of -modules. Then is said to be pure if it remains exact after tensoring with any -module. In other words, if is any -module, then
is a pure exact sequence.
From this definition, it is clear that is a pure subgroup of iff is a pure -submodule of .
Remark. is a pure submodule of over iff whenever a finite sum
where and implies that
for some . As a result, if is an ideal of , then the purity of in means that , which is a generalization of the second condition in the definition of a pure subgroup above.
|Date of creation||2013-03-22 14:57:47|
|Last modified on||2013-03-22 14:57:47|
|Last modified by||CWoo (3771)|
|Defines||pure exact sequence|